Hi, I've been stuck with this problem for a while:
if z=z(x,y), x=x(u,v) and y=y(u,v), prove that this equation is true:
\(\displaystyle \dfrac{\partial^2 z}{\partial u^2}\, =\, \dfrac{\partial^2 z}{\partial x^2}\, \left(\dfrac{\partial x}{\partial u}\right)^2\, +\, 2\, \dfrac{\partial^2 z}{\partial x \partial y}\, \dfrac{\partial x}{\partial u}\, \dfrac{\partial y}{\partial u}\, +\, \dfrac{\partial^2 z}{\partial y^2}\, \left(\dfrac{\partial y}{\partial u}\right)^2\, +\, \dfrac{\partial z}{\partial x}\, \dfrac{\partial^2 x}{\partial u^2}\, +\, \dfrac{\partial z}{\partial y}\, \dfrac{\partial^2 y}{\partial u^2}\)
Now, I understand that this is probably just a formula, and I get a result partially, save the part that begins with a 2 (the mixed partial derivative). Could anyone explain where it comes from?
Thank you.
if z=z(x,y), x=x(u,v) and y=y(u,v), prove that this equation is true:
\(\displaystyle \dfrac{\partial^2 z}{\partial u^2}\, =\, \dfrac{\partial^2 z}{\partial x^2}\, \left(\dfrac{\partial x}{\partial u}\right)^2\, +\, 2\, \dfrac{\partial^2 z}{\partial x \partial y}\, \dfrac{\partial x}{\partial u}\, \dfrac{\partial y}{\partial u}\, +\, \dfrac{\partial^2 z}{\partial y^2}\, \left(\dfrac{\partial y}{\partial u}\right)^2\, +\, \dfrac{\partial z}{\partial x}\, \dfrac{\partial^2 x}{\partial u^2}\, +\, \dfrac{\partial z}{\partial y}\, \dfrac{\partial^2 y}{\partial u^2}\)
Now, I understand that this is probably just a formula, and I get a result partially, save the part that begins with a 2 (the mixed partial derivative). Could anyone explain where it comes from?
Thank you.
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